In [1]}, a higher dimensional version of the SG theorem was proved, with lines replaced by flats.
This variant has applications in the study of algebraic circuits, and in particular in Polynomial Identity Testing (PIT) [2]}, [3]}, a central problem in algebraic complexity theory.
The works [2]}, [3]} use the higher dimensional Sylvester-Gallai theorems to bound the “rank" of certain types of depth three circuits.Algebraic circuits which compute polynomials that can be written as a sum of products of linear forms.
In simple terms, if the linear forms of a circuit satisfy the high dimensional SG condition, then in essence the polynomial being computed must depend on a constant number of variables, in which case it is easy to check whether the circuit is computing a non-zero polynomial.
